Asymptotic expected number of Nash equilibria of two-player normal form games

The formula given by McLennan [The mean number of real roots of a multihomogeneous system of polynomial equations, Amer. J. Math. 124 (2002) 49-73] is applied to the mean number of Nash equilibria of random two-player normal form games in which the two players have M and N pure strategies respectively. Holding M fixed while N → ∞, the expected number of Nash equilibria is approximately (√(π log N) / 2)(M-1)/√ M. Letting M = N → ∞, the expected number of Nash equilibria is exp(NM + O(log N)), where M ≈ 0.281644 is a constant, and almost all equilibria have each player assigning positive probability to approximately 31.5915 percent of her pure strategies. © 2004 Elsevier Inc. All rights reserved.